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Theorem 1 of [LS] Liebeck and Seitz - On the subgroup structure of exceptional groups says:

Let $X = X(q)$ be a quasisimple group of Lie type in characteristic $p$, and suppose that $X < G$, where $G$ is a simple adjoint algebraic group of exceptional type, also in characteristic $p$. Assume that [$q$ is large]. Then … there is a closed connected subgroup $\overline X$ of $G$ containing $X$, such that every $X$-invariant subspace of the Lie algebra $L(G)$ is also $\overline X$-invariant ….

[LS] emphasises that the condition is only on $q$, not on $p$; that is, it is OK for the characteristic $p$ to be small, as long as we take a large power of it (usually $q > 9$). Subgroups $X$ as described are called generic. The much more recent Litterick - On non-generic finite subgroups of exceptional algebraic groups justifies the restriction to non-generic groups by saying that the generic case is well understood, and points to [LS], which I take indirectly to be indicating that [LS] remains the state of the art.

What, if anything, is known about generic subgroups, in the above sense, for small $q$, especially in the spirit of the quoted Theorem 1?

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The state of the art for subgroups $\operatorname{PSL}_2(q)$, which constitute most of the cases where $q > 9$ in [LS], is now given by David Craven's Memoir Maximal $\operatorname{PSL}_2$ Subgroups of Exceptional Groups of Lie Type (arxiv version). For maximal subgroups, complete information for groups of type $F_4$ and $E_6$ is given in another preprint The Maximal Subgroups of the Exceptional Groups $F_4(q)$, $E_6(q)$ and $^2E_6(q)$ and Related Almost Simple Groups of Craven. The case where $X(q)$ has untwisted rank more than $\frac{1}{2}\operatorname{rank}(G)$ (but arbitrary $q$) is given by Liebeck and Seitz in Maximal Subgroups of Large Rank in Exceptional Groups of Lie Type.

In all, this leaves a very small number of subgroup types; I'm not sure anyone has worried about the remaining cases explicitly.

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  • $\begingroup$ Thank you! Given Liebeck and Seitz's work handling maximal positive-dimensional subgroups, which involves a lot of careful computations down to $q = 2$ and $q = 3$, which as I understand it was backed up by considerable WeightCompare-aided computation, I thought someone might have found a way to reduce these small cases to something that could have been handled by exhaustive search. Is the expectation that that's not possible, or just that no-one's been interested in doing it? $\endgroup$
    – LSpice
    Commented Mar 10, 2023 at 14:05
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    $\begingroup$ I would very much expect it to be do-able now, with a combination of the "feasible character" methods of my PhD thesis/Memoir and Craven's notion of "pressure" to bring in unipotent classes - In fact this is what most of the progress since then has done, more or less. I think it is mostly a lack of interest that hasn't closed the gap (caveat: I have been working on other things and may well have missed recent papers on this). $\endgroup$
    – Alastair Litterick
    Commented Mar 10, 2023 at 21:27

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